Hausdorff distances between couplings and optimal transportation

Abstract

We consider optimal transportation of measures on metric and topological spaces in the case where the cost function and marginal distributions depend on a parameter with values in a metric space. The Hausdorff distance between the sets of probability measures with prescribed marginals is estimated in terms of the distances between the marginals themselves. This estimate is used to prove the continuity of the cost of optimal transportation with respect to the parameter in the case of the continuous dependence of the cost function and marginal distributions on this parameter. Existence of approximate optimal plans continuous with respect to the parameter is established. It is shown that the optimal plan is continuous with respect to the parameter in the case of uniqueness. However, examples are constructed when there is no continuous selection of optimal plans. Another application of the estimate for the Hausdorff distance concerns discrete approximations of the transportation problem. Finally, a general result on the convergence of Monge optimal mappings is proved. Bibliography: 46 titles.